Finite approximation of free groups II: the Theorems of Ash, Herwig-Lascar and Ribes-Zalesskii -- revisited and strengthened

TL;DR

This paper explores the deep connections between three fundamental theorems in group theory, demonstrating their equivalence and strengthening their results through new constructions, all within the framework of finite approximation of free groups.

Contribution

It establishes the equivalence of Ash, Herwig-Lascar, and Ribes-Zalesskii theorems and introduces strengthened versions using novel group constructions.

Findings

The three theorems are mutually derivable and equivalent.

Strengthened theorems are substantially more powerful than classical results.

All results can be viewed as instances of finite approximation of free groups.

Abstract

Relations and interactions between the theorems of Ash, Herwig--Lascar and Ribes--Zalesskii are discussed and it is shown that these three theorems are equivalent in the sense that each of them can be derived from each other one. Some strengthenings of these theorems are obtained with the use of groups provided by a construction of the third author. Evidence is given that these strengthenings are substantially stronger than the classical results. Yet, it turns out that both kinds of results can be interpreted as different instances of the same common scheme, namely as \emph{finite approximation of free groups}.